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In Summary
Key Idea
• Rational numbers include integers, fractions, their decimal equivalents,
and their opposites.
Need to Know
• Rational numbers can be positive, negative, or zero.
• Every integer is a rational number because it can be written as the quotient
of two integers.
25
210
5
For example, some ways 25 can be written are 1 , 21, and 2 .
• Just as with fractions, there are many ways to write the same rational
number.
25
5
5
10
For example, 22.5 5 2 5 22 5 22 5 22.50 5 2 4 .
3
• Every rational number (except 0) has an opposite. For example, 22 4
3
and 2 4 are opposites, since they are both the same distance from 0 on a
number line.
Checking
1.
Write the following rational numbers as quotients of two integers.
1
6
3
b) 22 8
5
c) 27 6
a) 20.5
b) 22 4
c) 25 7
d) 7.2
2. Write the following rational numbers in decimal form.
1
a) 24
3.
Reading Strategy
Visualizing
3
d) 4 10
List three rational numbers between each pair.
2
4
1
a) 23 and 25
1
1
b) 22 and 4
c) 0.6 and 1 8
Use number lines to
visualize your answers
to question 3.
Practising
4.
5.
A.
22.2 and 21.5
B.
22.1 and 21.2
3
1
C. 22 and 21
4
2
29
3
D.
and 22
4
A.
NEL
⫺3
Y
⫺1
⫺2
0
Multiple choice. Which of these rational numbers are equivalent?
X: 24.2
6.
X
Multiple choice. Which values describe the positions of X and Y ?
X, Z, and W
Y: 22.4
B.
X and Z
42
Z: 210
C.
X and W
24
W: 10
D.
X and Y
Identify the values represented by B, D, K, M, and T as quotients
of two integers.
T
⫺4
M
⫺2
K
D
0
1.1 Interpreting Rational Numbers
B
2
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Write each as the quotient of two rational numbers.
1
2
a) 5.1
b) 24 5
c) 23.02
d) 9 3
8. Write each in decimal form.
2
2
19
213
a) 5 5
b) 27 4
c) 2 3
d) 25
7.
9.
Draw a number line. Mark integers from 210 to 2 on it. Estimate
and mark the location of each of these on your number line.
5
2
a) 22
c) 27.2
e) 23 3
g) 20.78
2
2
b) 13
d) 26.9
f) 3
h) 28.4
10. a) Describe a real-life situation that involves the number 21.2.
2
b) Describe a real-life situation that involves the number 23.
23
3
3
Explain why 24 5 4 5 24.
3
2
12. a) Name three fractions between and .
8
8
b) How would your answers in part a) help you name three
11.
2
Natural
numbers
Whole numbers
3
rational numbers between 28 and 28?
c) Are your answers in part a) rational numbers? Explain.
13. Agree or disagree with each. Explain why.
a) The opposite of every mixed number can be written as a
rational number in decimal form.
b) If one number is greater than another, so is its opposite.
14. The natural numbers are the numbers 1, 2, 3, 4,...
The whole numbers are the numbers 0, 1, 2, 3,...
Extend this Venn diagram to show the relationship between these
sets of numbers: integers, whole numbers, natural numbers, and
rational numbers.
Closing
15.
If you were describing rational numbers to someone without just
repeating the definition, what is the most important thing you could
say to help them quickly understand what rational numbers are?
Extending
16.
5
a) Explain why 9 can be written as the repeating decimal 0.555...
(or 0.5).
5
b) How would you write 29 as a decimal?
5
8
c) How would you write 290 as a repeating decimal?
d) How are your answers to parts b) and c) related?
17. Design an appropriate symbol for the term rational number.
Explain why your symbol is appropriate.
Chapter 1 Rational Numbers
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In Summary
Key Ideas
• Rationals can be compared in the same ways as integers and fractions.
• Negatives are always less than positives. A negative farther from 0 is
always less than a negative closer to 0.
Need to Know
• To compare rationals in fraction form, it helps to use mixed number
representations, equivalent fractions with common denominators or
common numerators, or benchmarks. For example,
1
1
23 2 . 24 3 since 23 . 24
3
7
24
2
1
2
35
21
21
25 . 28 since 240 . 240 or since 235 . 224
1
1
1
23 , 25 since 23 , 22 and 25 . 22
• You can also express all the rational numbers as decimals and compare
them in decimal form.
Checking
2
2
1.
List three rationals between 25 3 and 24 3.
2.
Explain why 23 4 , 23 8.
3.
You and your friends are standing at a point called 0. You use
positive numbers to describe going east and negative numbers to
describe going west. One friend moves to 13.2. Another moves to
22.3. A third moves to 20.9. Which of your friends moved the
least distance? In what direction?
3
1
Practising
4.
Use ., ,, or 5 to make each statement true. Explain how you
know parts b) and d) are true.
9
d) 24 2 ■ 22
b) 24.3 ■ 23.4
e) 5.6 ■ 5 5
2
2
c) 21 5 ■ 1 5
NEL
1
a) 0 ■ 20.5
3
3
f ) 22 10 ■ 2.3
1.2 Comparing and Ordering Rational Numbers
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5.
Replace the ■ with a value to make each statement true.
a) 22.3 , 2■.4
d) 1.9 , ■.4
b) 21.■ , 2■.8
e) 21.■ . 21.■
c) 20.■ , 0.■
f ) ■.8 . 4.■
6.
Order each from least to greatest.
1
1
2
a) 212.2, 214 2, 3 4, 24 3
2
b) 28.4, 210.2, 210.3, 8.8, 8 3
3
3 3 23
c) 8,28, 15, 15
3
d) 14.2, 22.9, 22 4, 10.1
7.
Draw X and Y on a number line to make both statements true.
X is between 22 and 23 but greater than 22.8.
Y is greater than X but less than 22.2.
8.
Multiple choice. Which number sentence describes how P
compares to Q?
P
Q
⫺6
A.
B.
9.
⫺5
25.6 , 24.8
25.3 , 24.4
⫺4
C.
D.
25.6 . 24.8
25.3 . 24.4
1
Multiple choice. Which correctly orders 18.1, 24.3, 24 3, 10.9,
24
and 2 5 from least to greatest?
1
24
A.
24.3, 24 3, 2 5 , 0.9, 8.1
B.
24 3, 24.3, 2 5 , 0.9, 8.1
1
24
24
1
24
1
C.
2 5 , 24 3, 24.3, 0.9, 8.1
D.
2 5 , 24 3, 24.3, 8.1, 0.9
10.
Earth scientists sometimes use negative values to describe the time
before a major earthquake. Suppose a scientist observed a tremor at
212.2 s and another tremor at 210.4 s. Which tremor occurred
first? How do you know?
11.
Why would you probably use a different strategy to compare 24.3
1
and 24 3 than to compare 18.1 and 24.3?
12.
a) Describe three different strategies you might use to order these
rational numbers.
11
22,
7
1
1
5
1
27.5, 25, 21 2, 22 2, 8, 100
b) Use your strategies to order the numbers from least to greatest.
14
Chapter 1 Rational Numbers
NEL
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Point Q, 21.25, is between point P at 21.2 and point R at 21.3.
RQP
⫺1.5 ⫺1.3 ⫺1.2
⫺1.25
⫺1
a) Copy the number line. Draw a point S between P and Q . Label
it with a value.
b) Draw a point T between P and S. Label it with a value.
c) Repeat two more times, each time drawing a new point between
P and the last point and labelling it with a value.
d) How can you be sure there is always another rational number
between two given ones? Use an example to explain.
e) What does this tell you about how many rational numbers there
must be?
14.
If a and b are positive numbers and a , b, how do 2a and 2b
compare? Explain why.
15.
Rational numbers can be written in either decimal form or fraction
form. Which form do you find easier to use when ordering rational
numbers? Use examples to justify your decision.
Closing
16.
Agree or disagree with the following statement and explain your
reasons. “If you can order integers and you can order fractions, you
have all the skills needed to order rational numbers.”
Extending
1
24
24
17.
Order these from least to greatest: 20.242 424..., 24, 2100, 298
18.
The same digit is substituted into each blank. The order from least
to greatest is
3
1
■
2
2■ 5, 27 ■, 24 11, 24 ■.
What could the digit be? Explain.
19.
NEL
Let x represent a value that could be anywhere between 22.4 and
23.8. Let y represent a value that could be anywhere between 24
and 23.5. About what fraction of the time will it be true that x , y ?
1.2 Comparing and Ordering Rational Numbers
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Checking
1.
Evaluate.
a) 24.2 1 (23.8)
2.
e) 23 2 Q24R
d) 2.5 2 5.6
f ) 23 5 2 1 3
1
b) 8 1 Q23R
7
c) 2 5 1 Q22R
2
1
4
3
4
2
You lose $1.20 on each share you own and then gain back $0.65.
Write the total loss on each share as a rational number.
Practising
3.
Estimate the sums or differences. Explain your thinking.
a) 3.64 2 72.9
d) 0.47 2 (221.6)
b) 212.2 2 (218.9) e) 3.42 2 (25.6) 1 11.3
c) 29.37 2 5.93
f ) 25.1 1 (25.82) 1 5.01
4.
Calculate exact answers for question 3.
5.
Multiple choice. Which sum or difference is about 116?
A. 22.3 2 18.4
C. 24.1 2 (219.8)
B. 14.1 1 (22.1)
D. 23.98 1 (28.9)
6.
Multiple choice. Yaroslav takes 4 h to cut his family’s front lawn
3
1
and 1 3 h to cut the back lawn. How much longer does it take
Yaroslav to cut the back lawn than the front?
A.
B.
1
13 h
C.
35 min
D.
45 min
7.
Consider these numbers: 24.2, 28.94, 25.362, 19.4, 11.205
Which two numbers have
a) a sum of 5.2?
c) a sum of 4.038?
b) a difference of 6.567? d) a difference of 23.578?
8.
Determine the missing digits for each. Use a calculator to help you.
a) 23.5■2 1 ■.42■ 5 1.846
b) 21■.382 2(4.17■) 1 8.■3 5 27.■27
c) 22.45■ 2(25.■63) 5 ■.705
d) 25.1■ 2(2■.8) 2 7.■ 5 29.21
9.
a) How could using the zero principle help you add 3.4 1 (28.9)?
b) Why would the zero principle not help you add 23.4 1 (28.9)?
What other strategy could you use instead?
10.
20
1
12 h
Chapter 1 Rational Numbers
Calculate. Show your work.
3
3
3
6
a) 28 1 1 4
c) 5 2 2
1
2
3
2
b) 25 2 1 2 3
d) 1 4 1 Q23 5R
2
1
e) 23 3 2 4 5
5
11
f ) 28 2 Q2 3 R
NEL
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The daily changes in price for a share during a week were 2$2.78,
2$5.45, $0.38, $1.38, and $2.12. The price of the share was $58.22
at the start of the week. What was the price at the end of the week?
How do you know that 22.3 2 Q23 4R is
1
a) greater than 22.3 1 Q23 4R ?
1
b) about 1?
Determine the value that makes each equation true.
3
3
a) 21 4 1 ■ 5 1
b) 21 4 2 ■ 5 1
James finished the Manitoba Marathon in a time of
3:57:53.3 (hours: minutes: seconds). The winner of the
marathon finished in a time of 2:25:55.6. Determine how much
longer James took to complete the marathon than the winner did.
15. Evaluate each expression for the given values.
a) x 2 y when x 5 24.1 and y 5 23.2
b) x 1 y 1 z when x 5 2.5, y 5 27.8, and z 5 24.1
14.
1
3
c) x 2 y when x 5 22 2 and y 5 23 4
1
1
d) x 1 y when x 5 21 2 and y 5 2 4
To recreate the work of the voyageurs during the fur trade, a relay
race was held on the Red River near St-Boniface, MB. Participants
canoed to specific points to find a message like those at right, which
led them to a fur cache. What rational number operations would
you use to determine each of the following?
a) the distance of the last leg
b) the total distance paddled
17. List two rational numbers a and b that are not integers and that
would make each statement true.
a) a 1 b is negative, but a 2 b is positive.
b) a 1 b is positive, but a 2 b is negative.
c) a 1 b and a 2 b are both negative.
18. Describe a real-world problem where you might calculate
23.2 2 (24.5). Solve your problem.
16.
Message 1: 1.5 km south
Message 2: 0.68 km north
Message 3: 2.3 km south
Message 4: north to the
starting point
Closing
19.
Describe a strategy for calculating the sum and a strategy for
calculating the difference of 23.4 and 15.005.
Extending
23
3
The sum of two rational numbers is 40. Their difference is 21 40.
What are the numbers?
21. The sum of two rationals is 17.4 less than the difference. What
could the rationals be?
20.
NEL
1.3 Adding and Subtracting Rational Numbers
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In Summary
Key Ideas
• Multiplying and dividing rational numbers in decimal form combines the
rules for multiplying and dividing positive decimals with the rules for
multiplying and dividing integers. For example,
(23.2) 4 1.2 5 2(3.2 4 1.2)
• Multiplying and dividing rational numbers in the form of fractions
combines the rules for multiplying and dividing positive fractions with
the rules for multiplying and dividing integers. For example,
5 4 3 Q22 3R 5 2Q 4 3 3R
3
1
23
7
Need to Know
• You can divide rational numbers in the form of fractions by using a
common denominator and dividing the numerators. For example,
12
3
12
15
225 4 5 5 225 4 25
• You can also divide by multiplying by the reciprocal. For example,
12
3
12
5
225 4 5 5 225 3 3
Checking
1.
Evaluate.
a) (22) (9.5)
c)
24
6
b) 7 3 25
d)
23
(28) 4 (0.5)
2
5
4 Q28R
5
23
5
5
2.
How much less is 4 4 6 than 4 3 6?
3.
A water tank lost 3 of its volume of water one day and then 2 of
what was left the next day. What rational number describes the
volume of water after the second day as compared to the original
volume?
1
1
Practising
4.
Calculate.
2
5
a) 23 3 8
2
25
b) 23 3 8
NEL
2
28
c) 3 3 5
5 2
d) 28 4 3
e)
f)
5
22
4 Q28R
3
2 5
4
3 8
1.4 Multiplying and Dividing Rational Numbers
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1
Multiple choice. Which expression is about 22?
A.
6.
2
1
23 3 8
B.
8
1
4 Q22R
9
C.
4
1
4 Q1 2R
5
D.
2
5
23 4 4
Multiple choice. Without evaluating, determine which expressions
3 5
have the same product as Q 4RQ 8R.
W: Q 24RQ 8R X: 2Q 4RQ28R Y: Q28RQ24R Z: Q 4 RQ 28R
A. X and Y
C. X, Y, and Z
B. X and Z
D. all of these expressions
23
Year
Temperature (8C)
2002
220.4
2003
27.6
2004
215.8
2005
29.3
2006
210.5
2007
216.3
5
3
5
3
5
23
5
■
2
7.
Use the numbers 21, 23, and 8 in the blanks so that ■ 3 ■ 3 has
a) the least possible value
b) the greatest possible value
8.
Consider the numbers 24.2, 21.3, 28.4, and 7.3.
a) Which two have a product of 35.28?
b) Which two have a quotient of about 21.75?
9.
The temperatures at Fort Nelson, BC, at 5:00 a.m. on December 25
from 2002 to 2007 are shown in the table. Determine the mean
temperature at 5:00 a.m. on December 25 for these years.
10.
Calculate. Show your work.
5
28
15
1
a) Q 212RQ 15 R
c) 16 4 Q21 24R
b) Q3 7RQ28 3R
6
Source: Environment Canada
1
2
7
d) 24 3 4 12
e) (23.2) 4 (28.4)
f ) 7.2 4 (20.6)
11.
A formula to convert temperatures between degrees Fahrenheit and
5
degrees Celsius is C 5 9 (F 2 32). Use this formula to convert the
following.
a) Miami, Florida’s record high of 98 °F to degrees Celsius
b) Anchorage, Alaska’s record low of 237 °F to degrees Celsius
c) 0 °C to degrees Fahrenheit
12.
Two tanks hold the same amount of water. Tank 1 loses 3 of its
1
volume. Tank 2 gains 4 of its volume. What is the final ratio of
water volume, comparing tank 1 to tank 2?
13.
An investment loses 2 of its value and then loses another 3 of the
new value.
a) What fraction of its original value is the final value?
2
1
2
b) Can you multiply 22 3 Q23R to calculate the answer to
1
2
part a)? Explain.
26
Chapter 1 Rational Numbers
NEL
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1
A pail of water has been sitting for a while, and 8 of the water has
evaporated.
a) What could 4 3 Q28R describe about this situation?
3
1
b) What could 28 4 Q24R describe about this situation?
1
1
12
15.
The product of two rationals is 225. What might their quotient be?
16.
The product of 24 and two other rationals is 4. The quotient of the
3
1
3
two other rationals is 24.
●
3
▲
1
24 3 ◆ 3 ■ 5 4
● ▲
3
4 ■ 5 24
◆
a) How do you know that one unknown rational is positive and
one is negative?
b) What could the unknown rationals be?
17.
Evaluate each expression for the given values. Use a calculator.
a) x 2 2y when x 5 29.78 and y 5 3.2
b) (x 1 y) (x 2 y) when x 5 2.5 and y 5 27.8
1
3
c) x(x 1 y) when x 5 22 2 and y 5 3 4
x
y
1
1
d) y 1 x when x 5 21 2 and y 5 2 4
Closing
18.
Create a brief “instruction manual” to help someone with the rules
for multiplying and dividing rational numbers.
Extending
19.
Calculate each product.
a) 29 3 0.2222...
b) 299 3 0.232 323...
20.
The product of a positive and a negative rational number is
2 greater than their sum. What could they be?
21.
The width of a rectangle is 4 of the length. If you increase the width
by 12 m and double the length, you obtain a perimeter of 60 m.
Determine the dimensions of the original rectangle.
NEL
1
1.4 Multiplying and Dividing Rational Numbers
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1
5
3
2
3
24 2 8 5 28 2 8
5
5 28
The new level is 8 of a pail lower than the original level.
Q:
How can you multiply and divide rational numbers to solve
problems?
A:
You multiply and divide rationals by combining the rules you
know for multiplying and dividing integers with those for
multiplying and dividing positive fractions or decimals.
Study
Aid
• See Lesson 1.4, Example 2.
• Try Mid-Chapter Review
questions 12, 13, and 14.
3
For example, suppose one investment loses 4 of its value. Then, it
5
loses 8 of that new value. What fraction of the value of the original
investment is the final value?
1
3
3
1
3 8 5 32 is the final value. That’s because you end up with 4 of
4
3
the value after the first loss and then 8 of that after the second loss.
For example, suppose you have two investments of equal value.
3
5
The first investment loses 4 of its value; the second loses 8 of its
value. How much more is the loss per dollar on the first investment
than on the second one?
24 4 Q28R 5 8 4 8
3
5
6
5
6
55
6
For each dollar lost on the second investment, you lose 5 of a
dollar, or $1.20, on the first investment.
Practice
Lesson 1.1
A
1.
What rational numbers describe the points A, B, and C ?
2.
Locate each rational on a number line from 210 to 110.
218
23
1
a) 22.6
b) 4
c) 23
d) 28 3
3.
On a February day, the daytime high temperature in Saddle Lake
First Nations Reserve, AB, was 24.5 °C. The temperature in
Portage la Prairie, MB, on the same day was 212.8 °C. Which
place was colder? Explain.
NEL
–5
B
C
0
Mid-Chapter Review
2
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Lesson 1.2
4.
Use . , , , or 5 to make true statements. Explain how you know
each statement is true.
2
5
2
5
1
9
2
3
a) 23 ■ 26 b) 3 ■ 8 c) 22 4 ■ 24 d) 25 ■ 210
5.
Write these rational numbers in order from least to greatest.
3 1
1
a) 25, 23, 21 3
c) 0.7, 20.3, 20.3
22
1 4
b) 5 , 22 5, 5
d) 0, 21.5, 22
3
1
6.
List four rational numbers between 28 and 22.
7.
A rational number of the form 6 is between 24 and 22. What is
the numerator of the rational number?
■
3
1
Lesson 1.3
8.
Calculate.
1
1
a) 24 2 53
5
b) 5 1 Q29R
c) 27.2 2 (24.8)
3
d) 28 1 Q23R
8
1
e) 23.5 1 (27.7)
f ) 27 14 1 Q24R
9. Kristen walked 5.7 km east and then 9.1 km west. How far east or
west was Kristen from her original position?
10.
1
1
3
1
The difference of two rational numbers is 5. The sum is 3. What are
the rational numbers?
Lesson 1.4
11.
The daily changes in selling price for shares in Robots Inc. during
a week were 2$4.50, 2$0.95, $0.25, 2$2.36, and 2$3.72. What
was the mean daily change in selling price for the share during
this week?
12.
a) Create two other expressions that give the same answer as
Q21 4RQ5 3R.
3
1
b) Describe a situation that Q21 4RQ5 3R might represent.
30
3
1
13.
Noah had 1.5 times as much savings as Kelly had debt. Then, Kelly
doubled her debt. What rational number division describes Noah’s
money as compared to Kelly’s?
14.
Create and solve your own problem involving multiplying or
dividing rational numbers.
Chapter 1 Rational Numbers
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In Summary
Key Idea
• The rules for order of operations with rationals are the same as with
integers:
• Do what is in brackets first.
• Multiply and divide from left to right.
• Add and subtract from left to right.
Checking
1.
2.
Which operation would you perform first in each situation?
a) 2.4 2 33.5 1 (22.9)4 3 (25)
b) (25.2 2 1.4) 4 (23) 3 (2.1)
Share values went up and down over the course of a week as shown
at right. What was the mean daily change?
Practising
3.
Which operation would you perform last in each situation?
a) 3.7 33.5 1 (22.9)4 4 32.5 1 (21.8)4
b) 8 4 (21.3) 1 (26.8 2 3.4)
2
5
3
Day
Share Value
1
10.005
2
20.135
3
20.115
4
20.12
5
10.05
9
c) 3 3 4 2 4 3 5
4.
Temperatures went up and down over the course of five days as follows:
24.2 °C, 21.4 °C, 11.9 °C, 13.7 °C, 21.8 °C
What was the mean daily change?
5.
Multiple choice. What is the value of
32.4 2 5.74 3 325.1 2 (21.8)4 1 0.2?
A. 11.09 B. 17.23 C. 17.83
D. 16.23
6.
Multiple choice. What is the value of
20.7 2 0.3 4 (20.15) 3 0.2 1 2?
A. 0.4
B. 1.7
C. 20.46 D. 21.6090909...
7.
8.
NEL
Mika calculated Q24 R 1Q23 R 4 3 as 24 4. Is Mika correct? Explain.
3
2
1
1
Evaluate.
2
1
3
a) 23 3x 4 y when x 5 2 2 and y 5 23 4
5
3
7
b) 2x 2 3 y when x 5 21 4 and y 5 210
1.5 Order of Operations with Rational Numbers
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9.
10.
Page 34
The temperature on Friday went up 3.1 °C from Thursday. After a
loss of 21.2 °C on Saturday, the temperature doubled on Sunday.
How did the temperature on Sunday compare to the temperature
on Thursday?
For which calculations would you not need to know the order of
operations rules?
a) Q23R 2 8 2 Q25R
2
5
4
b) Q23R 3 8 2 Q25R
2
5
Q23R 2 8 2 Q25R 3 2
2
c)
5
4
d) Q23R 4 8 3 Q25R
4
2
5
4
11.
a) Create an expression, involving both positive and negative
rational numbers that are not integers, for which you would
need to know the order of operations to calculate it correctly.
Explain why you would need to know the order of operation
rules for your expression.
b) Check to see if your calculator follows those rules for order of
operations when you enter rational numbers. How do you know?
12.
During the last heat for the 800 m run at the Jeux de la Francophonie
1
in Edmonton, Xavier gained 4 s over his nearest opponent during the
2
first 100 m. He then lost 3 s during the second 100 m, lost a further
1
1
s during the third 100 m, but gained 2 s during the fourth 100 m.
8
How much time will Xavier have to gain on his opponent during the
last 400 m to win the heat?
Closing
13.
Why is it essential that the rules for order of operations for rational
numbers be the same as the order of operations for integers?
Extending
14.
Create an expression involving rational numbers where you perform
an addition before you perform any multiplications or divisions.
Calculate the value of that expression.
15.
Where would you add brackets in each expression to make it true?
1
a) 3 2 1 4 4 0.75 1 (28.1) 5 1.9
1
b) 21.2 1 23 4 1.5 1 1.5 2 5 3 13 5 24
16.
2
You subtract a rational number from 23, double the answer, and
1
then divide by 24. The result is 8. What was the rational number?
34
Chapter 1 Rational Numbers
NEL
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Page 39
Checking
1.
5
You subtract a rational number from its triple. The result is 28.
What is the rational number?
Practising
2.
The mean of five numbers is 25. The sum of the positive numbers
in the set is 37 greater than the sum of the negative numbers in the
set. What could the numbers be?
3.
The number of kilometres in the perimeter of this park is 8 greater
than the number of square kilometres in its area. What is the width?
Use a calculator to help.
4.
The sum of three rationals is 0. Two have denominators of 8. The
1
product of two of them is 16. The quotient of two of them is 25.
What are the rationals?
5.
The product of two opposites is 21.8225. What are the numbers?
Use a calculator to help you guess and test.
6.
The sum of five numbers is 24. The numbers include two pairs of
opposites. The quotient of two values is 2. The quotient of two
? km
6.0 km
1
3
different values is 24. What are the values?
7.
8.
A share price increased in value one day, doubled that increase
the next day, and then decreased $0.12 in value the following day.
If the total change was $0.33, by how much did the share go up the
first day?
A rational number with a denominator of 9 is divided by Q23R. The
2
4
1
5
result is multiplied by 5 and then 26 is added. The final value is 10.
What was the original rational?
9.
1
Two rational numbers are added. The sum is 4 more than the
product. What are the rational numbers?
Closing
10.
Why is guess and test a good strategy to use for questions like
question 3? Why might you also work backward?
Extending
11.
NEL
Create a problem involving rational numbers that would be
reasonable to solve with a guess-and-test strategy. Then, solve it
using that strategy.
1.7 Solve Problems Using Guess and Test
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Chapter Self-Test
CHAPTER 1
1.
24
Multiple choice. Which value is equivalent to 25?
A.
2.
Page 41
24
5
B.
4
25
C.
4
25
D.
4
5
Multiple choice. Which set of numbers is arranged in order from
least to greatest? Explain.
11 211
2
2
11 211
A.
2 5 , 25 , 22 5
C.
22 5,2 5 , 25
B.
211
11
2
, 2 5 , 22 5
25
D.
22 5, 25 , 2 5
2 211
11
1
3.
Which value is farther from zero: 24 3 or 4.3? Explain.
4.
List three rationals between 23 4 and 2 8 .
5.
Calculate. Show your work.
24
1
4 8
5
a) 7 2 Q22 2R
d) Q25RQ 3RQ26R
1
2
3
b) 22 3 4 4
1
24
c) 23 5 1 Q 3 R 1 4.5
27
e) 3.2(24.1) 4 (2 1 3.4) 3 3
f ) 5 3 2 Q2 3 4 2R 1 3 4 2
1
1
8
6.
The high temperatures, in degrees Celsius, for a city during a
five-day period increased or decreased as follows:
14.2°, 11.7°, 211.7°, 22.3°, 15.2°
a) What was the total increase or decrease from the start of the
period until the end?
b) What was the average daily change in temperature?
7.
The height of the water level in a tank dropped by 4. Katia used
1
3
of what was left. What rational number describes the final
5
height as a fraction of the original height?
8.
The following are true about two rational numbers.
• The product is greater than the quotient.
• The sum is less than the difference.
List three things that you know about the numbers and explain
how you know those things.
WHAT DO You Think Now?
Revisit What Do You Think? on page 3. How have your answers and
explanations changed?
NEL
Chapter Self-Test
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Page 43
Practice
Lesson 1.1
1.
Locate each value on a number line.
24
a) 22.6
b) 2 5
229
231
2.
Which rational is between 210 and 29: 3 or 3 ? How do you
know?
3.
Write each as a quotient of integers.
4
a) 24.2
3
c) 28
b) 15
2
4.
Describe a situation that the number 213 might represent.
5.
What is wrong with this Venn diagram?
Integers
Rational
numbers
Lesson 1.2
6.
Order from least to greatest.
28
1
a) 25.1, 0.3, 3 , 1.2, 25
7.
2 28 4
1
List three rational numbers between each pair of rationals.
1
3
a) 24 3 and 24 4
8.
3
b) 5, 23, 9 , 7, 24
1
b) 25.01 and 25.006
4
2
c) 25 and 23
8
Explain why 22 . 22 even though 1 , 8.
Lesson 1.3
9.
2
Ann started gym 3 h before lunch. Her art class began an hour and
3
a half after lunch. Lunch lasted 4 h. Use a rational expression to tell
how many hours before art class her gym class started.
NEL
Chapter Review
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10.
Page 44
Estimate. Show your reasoning.
a) 23 2 2 Q28 4R
8
17
b) 3 1 Q2 5 R
1
3
c) 23.7 1 (217.1)
d) 3 1 Q2 5 R
2
16
11.
Calculate the sums and differences in question 10. Show your work.
12.
A share price increased by $0.05 one day, decreased by $0.02 the
next day, and decreased again by $0.01 the following day. What was
the total change?
13.
The sum of two rational numbers is 22. The difference is 210.
What are the two rational numbers?
1
11
Lesson 1.4
14.
Calculate. Show your work.
a) 22 Q25R
5
4
6
e) Q25R 4 Q23R
d) 7 4 Q214R
f ) (23) 4 Q21 3R
2
b) 3 Q25RQ23R
2
c) a13 b a29 b
5
2
4
9
6
2
2
15.
One share lost $0.25. Another share lost $0.03. What is the ratio of
the losses? Write the ratio as a rational number.
16.
The quotient of two rationals is 21.5. The product is 232.
a) What are the rationals?
b) How do you know there has to be another possible answer?
3
Lesson 1.5
17.
Calculate. Show your work.
2
23
1
1
22
12
a) 5 4 Q 5 1 10R
c) Q 8 1 3 R 3 13
5
2
3
1
21
3
b) 26 1 Q23R 4 4
d) 212 1 22 2 Q25R
18.
Aaron calculated 26.2 4 (3.1 1 1.9) 3 (22) as 29.8. Is this
correct? Explain.
19.
Use a calculator to determine how much less Q24 1 5R 4 3 is than
3
2
Q24 1 5R 3 3.
3
2
Lesson 1.6
20.
Determine two rational numbers a and b so that
a 3 b . a 4 b . a 2 b . a 1 b.
Lesson 1.7
21.
44
Chapter 1 Rational Numbers
The sum of three numbers is 1. One number is (22) times another.
The quotient of another pair of the numbers is 4. What are the
numbers? Explain.
NEL